Recently, the bounded, compact and Hilbert-Schmidt difference of composition operators on the Bergman spaces over the half-plane are characterized in (Choe et al. in Trans. Am. Math. Soc., 2016, in press). Motivated by this, we give a sufficient condition when two composition operators $C_{\varphi}$
 and $C_{\psi}$
 are in the same path component under the operator norm topology and show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if $C_{\varphi_{1}}$
 , $C_{\varphi_{2}}$
 , and $C_{\varphi_{3}}$
 are distinct and bounded, then $(C_{\varphi _{1}}-C_{\varphi_{2}})-(C_{\varphi_{3}}-C_{\varphi_{1}})$
 is compact if and only if both $C_{\varphi_{1}}-C_{\varphi_{2}}$
 and $C_{\varphi _{1}}-C_{\varphi_{3}}$
 are compact on weighted Bergman spaces over the half-plane. Moreover, we prove the strong continuity of composition operators semigroup induced by a one-parameter semigroup of holomorphic self-maps of half-plane.

/pdf/difference-of-composition-operators-on-weighted-bergman-x5u24va5na.pdf

Difference of composition operators on weighted Bergman spaces over the half-plane

Abstract We characterize the membership in the Schatten ideals $\mathcal {S}_p$ , $0&lt;p&lt;\infty $ , of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given. 

Composition operators on weighted analytic spaces

Difference of composition operators on the weighted Bergman spaces over the half-plane

Fock Projections on Mixed Norm Spaces

Ahlfors–David Regular Sets, Point Spectrum and Dirichlet Spaces

Littlewood–Paley estimates with applications to Toeplitz and integration operators on weighted Bergman spaces

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Littlewood–Paley estimates with applications to Toeplitz and integration operators on weighted Bergman spaces

Composition operators on weighted analytic spaces